Tertiary, quaternary and higher order states in the sequence of bifurcations approach (SBA) to turbulence for laterally heated shear flows within a rectangular tube

• Research highlight 1 - Application of the fully deterministic Sequence of Bifurcations Approach • Research highlight 2 - Vanishing flow through Lateral Section via imposition of a Constant Flux constraint • Research highlight 3 - The remote ends of the rectangular tube are assumed to be closed/periodic • Research highlight 4 - The large aspect ratio has potential to a variety of engineering applications • The tertiary OBV oscillatory states of the laterally heated rectangular tube in the SBA for P r = 0 , G r = 710 . The suharmonic state is a pulsating flow and it is depicted in a) for its azimuthal velocity and c) vorticity profiles. The equivalent superharmonic state is not pulsating and is depicted in b) for its azimuthal velocity and d) vorticity profiles. Here the wavenumbers are ( α , β = 1.34 , 0.7 ) for the subharmonic and ( α , β = 1.34 , 1.34 ) for the superharmonic state respectively. We consider a vertical rectangular tube of large aspect ratio with side-wall heating in order to mimic realistic experimental conditions. We therefore impose the condition that across any lateral cross-section of the rectangular tube the fluid flow vanishes. We find through our numerical analysis that oscillatory modes yield critical conditions and offer therefore sequential bifurcations that lead to the turbulent regime. Although the linear stability analysis is the same as the case where the imposed constant flux condition is absent, the corresponding nonlinear regime displays fundamentally different characteristics to the open narrow channel case. Here we focus on the sequence of bifurcations approach of a fluid enclosed in a rectangular tube, aligning with engineering applications. We additionally assume the limit of small Prandtl number and thus the effects caused by temperature perturbations are ignorable. Finally we identify the oscillatory states that lead to turbulence as the Grashof number increases up to the value 1000. Our fully nonlinear numerical analysis shows that all bifurcations are supercritical and here we concentrate on the critical axial wavenumber of the linear stability analysis of the laminar flow and its pairing with a specific azimuthal wavenumber.